Math CentralQuandaries & Queries


Question from Belle:

Determine the equation of the circle which has its center at (3,1) and tangent 3x-4y+5=0

Hi Belle,

Suppose that $(c,d)$ is the point of tangency of the line $3x - 4y + 5 = 0$ and the circle. Since you know the center of the circle if you can find $c$ and $d$ you can write the equation of the circle.

I would first write the equation of the tangent line in the form

\[y = mx + b\]

and from this read off the slope, $m$ of the tangent line. The line segment joining the center of the circle to $(c,d)$ is a radius of the circle and hence this lie is perpendicular to the tangent line. This gives you an equation relating $c$ and $d.$

Do you know another equation relating $c$ and $d?$

Solve the two equations for $c$ and $d.$


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